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A semiprime PI-ring having a faithful module with Krull dimension is a Goldie ring. (English. Russian original) Zbl 0833.16023
Russ. Math. Surv. 48, No. 6, 158 (1993); translation from Usp. Mat. Nauk 48, No. 6(294), 141-142 (1993).
From the introduction: It was proved by S. A. Amitsur and L. W. Small [J. Algebra 133, 244-248 (1990; Zbl 0725.16018), Proposition 3] that if $$R$$ is a semiprime PI-ring and $$M$$ is a faithful Noetherian left $$R$$-module, then $$R$$ is a left Noetherian ring and $$\text{K dim}(R)=\text{K dim}(M)$$. The assertion that $$R$$ is Noetherian was also obtained independently by the second author. In this note we consider a more general class of modules, namely those having Krull dimension in the sense of R. Gordon and J. C. Robson [Mem. Am. Math. Soc. 133 (1973; Zbl 0269.16017)]; Theorem. A semiprime PI-ring having a faithful left module with Krull dimension is a Goldie ring.

##### MSC:
 16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16N60 Prime and semiprime associative rings
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